Polynomialit{\'e} d'anneaux de repr{\'e}sentations modulaires projectives

TL;DR

This paper proves that the Grothendieck group of finite type projective modular representations of symmetric groups and their wreath products forms a polynomial ring, revealing a new algebraic structure in modular representation theory.

Contribution

It establishes that the graded ring formed by these representations is a polynomial ring, a novel result in the study of modular representation theory of symmetric groups.

Findings

The Grothendieck group of projective modular representations is a polynomial ring.

The structure holds for symmetric groups and their wreath products.

Induction defines the product in the graded ring.

Abstract

Consider the Grothendieck group of finite type projective modular representations of the symmetric groups on n letters, or more generally, of its wreath product with a finite group. They form a graded group, with a product defined using induction. We show that the resulting graded ring is a polynomial ring.